Positional number systems

Our decimal number system is known as a positional number system,
because the value of the number depends on the position of the digits.
For example, the number 123 has a very different value than the
number 321, although the same digits are used in both numbers.

(Although we are accustomed to our decimal number system, which is positional,
other ancient number systems, such as the Egyptian number system were not
positional, but rather used many additional symbols to represent larger
values.)

In a positional number system, the value of each digit is determined
by which place it appears in the full number. The lowest place value is
the rightmost position, and each successive position to the left has a
higher place value.

In our decimal number system, the rightmost position represents the
"ones" column, the next position represents the "tens" column, the next
position represents "hundreds", etc. Therefore, the number 123 represents
1 hundred and 2 tens and 3 ones, whereas the number
321 represents 3 hundreds and 2 tens and 1
one.

The values of each position correspond to powers of the base of the
number system. So for our decimal number system, which uses base 10,
the place values correspond to powers of 10:

...

1000

100

10

1

...

10^3

10^2

10^1

10^0

Converting from other number bases to decimal

Other number systems use different bases. The binary number system
uses base 2, so the place values of the digits
of a binary number correspond to powers of 2.
For example, the value of the binary number 10011
is determined by computing the place value of each of the digits of the
number:

1

0

0

1

1

the binary number

2^4

2^3

2^2

2^1

2^0

place values

So the binary number 10011 represents the
value

(1 * 2^4)

+

(0 * 2^3)

+

(0 * 2^2)

+

(1 * 2^1)

+

(1 * 2^0)

=

16

+

0

+

0

+

2

+

1

=

19

The same principle applies to any number base. For example, the number
2132 base 5 corresponds
to

2

1

3

2

number in base 5

5^3

5^2

5^1

5^0

place values

So the value of the number is

(2 * 5^3)

+

(1 * 5^2)

+

(3 * 5^1)

+

(2 * 5^0)

=

(2 * 125)

+

(1 * 25)

+

(3 * 5)

+

(2 * 1)

=

250

+

25

+

15

+

2

=

292

Converting from decimal to other number bases

In order to convert a decimal number into its representation in a different
number base, we have to be able to express the number in terms of powers
of the other base. For example, if we wish to convert the decimal number
100 to base 4, we must figure out how
to express 100 as the sum of powers of 4.

100

=

(1 * 64)

+

(2 * 16)

+

(1 * 4)

+

(0 * 1)

=

(1 * 4^3)

+

(2 * 4^2)

+

(1 * 4^1)

+

(0 * 4^0)

Then we use the coefficients of the powers of 4
to form the number as represented in base 4:

100

=

1 2 1 0

base 4

One way to do this is to repeatedly divide the decimal number by the
base in which it is to be converted, until the quotient becomes zero. As
the number is divided, the remainders - in reverse order - form the digits
of the number in the other base.

Example: Convert the decimal number 82 to base
6:

82/6

=

13

remainder 4

13/6

=

2

remainder 1

2/6

=

0

remainder 2

The answer is formed by taking the remainders in reverse order:
2 1 4 base 6